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Re: Bowditch Table 15
From: George Huxtable
Date: 2005 Jan 27, 00:07 +0000

```Jim Thompson writes-

>I searched through the rest of Bowditch 2002 for more references to the
>Table 15 procedure, and found this odd paraphraph in Chapter 22, Article
>2202:
>
>"Distance by vertical angle between the waterline and the top of an object
>is computed by solving the right triangle formed between the observer, the
>top of the object, and the waterline of the object by simple trigonometry.
>This assumes that the observer is at sea level, the Earth is flat between
>observer and object, there is no refraction, and the object and its
>waterline form a right angle. For most cases of practical significance,
>these assumptions produce no large errors.
>D = sqrt[(tan^2a/.0002419^2) + ((H-h)/0.7349)] - (tan a/.002419)
>where D is the distance in nautical miles, a is the corrected vertical
>angle, H is the height of the top of the object above sea level, and h is
>the observer's height of eye in feet. The constants (0.0002419 and 0.7349)
>account for refraction.".
>
>I don't see why the Table 15 equation is given in this paragraph.  Looks to
>me like a chunk of text is missing between the end of "...produce no large
>errors" and the equation, and that in fact the second half, which contains
>the equation, belongs to a missing title that should describe "Distance by
>Vertical Angle Measured Between Sea Horizon and Top of Object Beyond Sea
>Horizon".  Can someone check an older Bowditch?

================

Response from George.

Quite astounding! Bowditch is compounding his errors even further!

My edition, which is 1977 for vol 1, and 1981 for vol 2, has quite
different text, at section 503 in vol 2.

It is indeed quite reasonable to state, as Jim quotes from section 2202 of
the 2002 edition:

>"Distance by vertical angle between the waterline and the top of an object
>is computed by solving the right triangle formed between the observer, the
>top of the object, and the waterline of the object by simple trigonometry.
>This assumes that the observer is at sea level, the Earth is flat between
>observer and object, there is no refraction, and the object and its
>waterline form a right angle. For most cases of practical significance,
>these assumptions produce no large errors."

=============
In fact, working on those simplifying assumptions, the "simple
trigonometry" would be to use

distance in feet = height in feet / tan angle

or distance in miles = height in feet / (6080 tan angle)

or (as near as dammit) for 106 ft height at a mile distant you will see an
angle of 1 degree. Smaller angle, then proportionately bigger distance.

In my edition, such problems are assisted by using table 41, "distance by
vertical angle; measured between waterline at object and top of object".
Does the newer edition carry that table, perhaps with a different number?
==============

But then, to follow that text directly with the formula for Table 15,
without further comment, is just crazy. Because that formula DOES allow for
the fact that the Earth isn't flat, and it DOES allow for refraction!
That's its whole point.

And notice the formula as Jim transcribed it. It's wrong, but differently
wrong compared to everything we've seen before.

D = sqrt[(tan^2a/.0002419^2) + ((H-h)/0.7349)] - (tan a/.002419)

In the third term, the constant divisor, of .002419, should have been .0002419.

For the first time, however, we see Bowditch putting the square-root
brackets in the right place, so that the square root embraces the first two
terms, and not the third.

In the earlier edition, in which it was Table 9, the explanation wrongly
put all three terms within the square root, and also made the same error as
above in the denominator of the last term.

In the explanation of Table 15 in the later edition, the denominator error
was corrected, but the square-root wrongly embraced just the first term of
the three.

So, in three tries, Bowditch has provided three different formulae for his
own table, NOT ONE of which is correct.

Assuming, that is, that the consensus Nav-l expression is indeed the
correct one, being-

d = sqr{(tan A / .0002419 )^2 + ((H-h) / .7349)} - (tan A / .0002419)

George.

================================================================
contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
================================================================

```
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